## Abstract

For any S ∩ Z we say that a graph G has the S-property if there exists an S-edge-weighting w: E(G) → S such that for any pair of adjacent vertices u, v we have Σ eϵ E(v) w(e) Σ eϵ E(u) w(e), where E(v) and E(u) are the sets of edges incident to v and u, respectively. This work focuses on {a, a + 2}-edge-weightings where a ϵ Z is odd. We show that a 2-connected bipartite graph has the {a, a + 2}-property if and only if it is not a so-called odd multi-cactus. In the case of trees, we show that only one case is pathological. That is, we show that all trees have the {a, a + 2}-property for odd a = -1, while there is an easy characterization of trees without the {-1, 1}-property.

Original language | English |
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Journal | Discussiones Mathematicae. Graph Theory |

Volume | 42 |

Issue number | 1 |

Pages (from-to) | 159–185 |

ISSN | 1234-3099 |

DOIs | |

Publication status | Published - 2022 |

## Keywords

- Neighbour-sum-distinguishing edge-weightings
- Bipartite graphs
- Odd weights
- 1-2-3 conjecture

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